Simplify the following by factorization and express in exponential form: 1. 135 multiplied by 375. Hi friends, can you please provide the answer? Thanks.
The Correct Answer and Explanation is:
To simplify 135 × 375 by factorization and express the result in exponential form, we begin with prime factorization of both numbers:
Step 1: Prime Factorization
135
= 5 × 27
= 5 × 3 × 9
= 5 × 3 × 3 × 3
= 3³ × 5
375
= 5 × 75
= 5 × 5 × 15
= 5 × 5 × 5 × 3
= 3 × 5³
Step 2: Multiply using Exponents
Now multiply:
135 × 375
= (3³ × 5) × (3 × 5³)
Use exponent rules:
- Multiply like bases by adding exponents:
3³ × 3¹ = 3⁴
5¹ × 5³ = 5⁴
So,
135 × 375 = 3⁴ × 5⁴
This is the simplified exponential form.
Step 3: Final Answer
✅ Answer: 3⁴ × 5⁴
Explanation
To simplify the product of 135 and 375 by factorization and express the result in exponential form, we first break each number into its prime factors. Prime factorization is the process of expressing a number as a product of its prime numbers. For example, 135 can be written as 3 × 3 × 3 × 5, which gives 3 raised to the power of 3 and 5 to the power of 1. In exponential form, we write this as 3³ × 5.
Similarly, 375 can be factorized as 3 × 5 × 5 × 5, which equals 3 × 5³.
Next, we apply the rules of exponents. When multiplying powers with the same base, we add the exponents. That means 3³ × 3 = 3⁴, and 5 × 5³ = 5⁴. So the product becomes 3⁴ × 5⁴.
This method not only simplifies the multiplication but also shows the structure of the number clearly. Exponential form is useful because it makes it easy to work with large numbers, especially in algebra and higher mathematics. Instead of multiplying the numbers directly to get 50,625 (which is the actual product), we express the number as a product of its prime factors raised to powers.
Therefore, the final answer in exponential form is 3⁴ × 5⁴.
